Explanation of the manuscript's Section 10: compact forcing and whole-space breakdown (pages 116 to 126)

Section 10, explained: Compact forcing and whole-space breakdown (pages 116 to 126)

The problem this section is handed

Section 9 ends with local fields u_loc, p_loc [p. 114], defined only on Ω∗ = {τ > 0, q < q∗}, where τ = 1 − t and q is the concentration scale [p. 7, p. 15]. Theorem 3.1 gives them four properties [pp. 15 to 16]: a representation u_loc = curl A + B e_θ, smooth at the axis; derivative bounds up to τ = 0 where q stays between positive constants below q∗; a residual that is flat (every derivative O(q^N) at bounded X as q ↓ 0) and exactly zero for X ≥ X_ext, where A = 0 and the flow is an explicit heat swirl K e_θ; and swirl growth τ^{−A} (exponent A = 1/2 + h, not the potential) along one path.

Theorem 1.1 needs five things these fields lack [p. 1]: compact support for velocity and pressure; zero initial velocity; a force in C_c^∞(R³ × (0, ∞)), smooth through t = 1 and beyond; a uniform energy bound; and a comparison showing that no global smooth solution with the same force and datum has bounded energy. Their domain stops at τ < q∗ (our reading, since τ ≤ q), their residual exists only for t < 1, and one construction says nothing about competitors. Section 10 supplies all five [p. 117].

What a person would try first, and why it fails

Cut the velocity. Multiply u_loc and p_loc by a bump equal to one near (0, 1). The divergence becomes ∇χ · u_loc, nonzero where the bump varies; projecting back onto divergence-free fields is nonlocal and loses compact support (our reading). The paper signals this by requiring the cutoff to act before a curl [p. 16, p. 117].

Trust flatness at t = 1. Take the residual as the force and rely on (3.4). But flatness holds at bounded X as q ↓ 0, near the singular point, while the cutoff creates terms such as p_loc ∇c in a transition region that reaches the terminal slice [p. 16]. The bounds available there depend on a positive lower bound for q [p. 119], and the paper names where that fails: positive radius with q → 0, on the plane z = 0 [p. 16, p. 119].

Compare by the global energy identity. To get nonexistence, show that any smooth competitor v with the same data equals u, via the energy identity for w = v − u. On R³ that needs decay of v, its derivatives and its pressure at infinity, and the paper assumes none [p. 121]. Some condition at infinity is unavoidable: with zero force and datum, a uniform velocity a(t), a(0) = 0, driven by the pressure −a′(t) · x, is a smooth nonzero solution (our reading).

The idea, motivated

Cut the potential. The construction already carries u_loc = curl A + B e_θ [p. 116], so cut the potentials: u = curl(cA) + cB e_θ, p = c p_loc (10.4). Both pieces are divergence-free, the second because cB is independent of θ, which is why the spatial cutoff is axisymmetric [p. 118] (M10.2). The product rule u = c u_loc + ∇c × A shows the price, an extra term wherever c varies [p. 118].

Which potential. So A must be controlled wherever c varies, including the axis and, as will emerge, the plane z = 0 near t = 1. The section checks two properties of its representative (9.21) [p. 115, p. 117] (M10.1). Near the axis the base streamfunction is S = r² a(r², z, t), so (S/r) e_θ = a(−x₂, x₁, 0) is smooth in Cartesian coordinates. And A = 0 for X ≥ X_ext: each streamfunction integrates an axial profile U_n outward from the axis (10.1) and returns to zero past its support because the total axial flux ∫_0^∞ U_n dX vanishes, a condition imposed in Sections 4 and 5 [p. 33, p. 49, p. 117]; wave and mean potentials vanish with their sources [p. 117]. Only the swirl B = K survives in the exterior [p. 116].

Where to cut. Ω∗ is described through q, defined by τ = q(1 − η²), z = q^D η, X = r²/(2q) [p. 7]. So q depends on (z, t) only, Ω∗ contains every radius, and the cutoff's radius r_0 is free [p. 118]. It remains to bound q by τ and |z| alone: if 1 − η² ≥ 1/2 then q ≤ 2τ, and otherwise q ≤ (√2|z|)^{1/D}, so q ≤ C_0(τ + |z|^{1/D}) uniformly in r (10.3) [p. 118]. Only now is the cutoff defined: c = χ_x χ_t, an axisymmetric spatial bump of radius r_0 and half-height z_0 times a time bump switched on after 1 − τ_0, with τ_0, z_0 so small that c lives in q < q∗/2, a fixed margin inside Ω∗; the time bump gives the zero datum (10.2) [p. 118]. The force is the residual, f = R(u, p) (10.5), so the equation holds by construction; the paper notes that f need not be divergence-free [p. 118], the price of cutting the pressure locally (our reading).

Where the force could fail at t = 1. Repair the second attempt: where does q vanish at τ = 0? If z ≠ 0, then q ≥ |z|^{1/D} > 0, Theorem 3.1(ii) bounds one more time derivative, and the fundamental theorem of calculus makes every derivative uniformly Cauchy [p. 119] (M10.3). If z = 0, then η = 0 and q = τ [p. 116]: q → 0 on the whole plane, not only at the origin. At fixed r > 0, X = r²/(2τ) then tends to infinity, so the point enters the heat exterior, where A = 0 and the flow is explicit, K = c∞ s^{−A} H(2τ/s) with s = r²/2 (10.7); derivatives of K and of its centrifugal pressure stay bounded down to τ = 0 (10.8) [p. 119, p. 138]. The uncut exterior solves the equations exactly, so only cutoff terms built from K and p_loc survive there, and they have limits [p. 119], a use the physical description announced [p. 6].

At the origin the individual terms diverge, but c = 1, so f is the local residual and (3.4) with (10.3) bounds every derivative by C(τ + |z|^{1/D})^N (10.9) [p. 119] (M10.4). A small ball, a late time and a finite cover then give uniform limits F_j of ∂_t^j f: smooth, supported in K, flat at the origin (10.6), and compatible (10.10), hence one-sided time Taylor data at t = 1 [pp. 118 to 120].

Past t = 1. Extension by zero would in general jump, and the Taylor series Σ σ^j F_j/j! may diverge (our reading). The repair damps term j by χ_0(b_j σ), σ = t − 1 (10.11) (M10.5): on its support σ ≤ 1/b_j, so each derivative costs a factor b_j and the term is O(b_j^{m−j}) (10.12); fast-growing b_j make it at most 2^{−j} in all norms of order up to ⌊j/2⌋. Every derivative then converges, the Taylor data are reproduced, and the support is K × [0, 2] [p. 120]. This is the classical Borel construction, unnamed in the paper; past t = 1 the force is one continuation among many (our reading).

Energy from the equation. The heuristic E_core ≍ τ^{1/2−3h} covers only the leading core [p. 16]; the paper reads the bound off the equation instead [p. 17] (M10.6). Compact support makes the energy identity (10.14) exact, and it bounds energy plus dissipation by F(t)², F(t) = ∫_0^t ‖f‖_2, finite since f is smooth (10.13) [p. 121].

From construction to nonexistence. Repair the third attempt. The paper assumes only v ∈ L^∞([0, T]; L²), T < 1 [p. 121], and spends it first on the pressure (M10.7). The force cancels in the difference equation (10.15), and the natural pressure is π∗ = Σ R_i R_j g_ij (Riesz transforms, g = v ⊗ v − u ⊗ u) (10.16) [p. 121]. The actual pressure difference π has the same Laplacian; averaged in time, ∇π − ∇π∗ lies in H^{−3}, because w ∈ L² and g ∈ L¹, and a harmonic element of H^{−3} vanishes, its Fourier transform being supported at the origin [p. 122]. The uniform flow fails here: its pressure gradient is a Fourier point mass, its energy infinite (our reading).

Boundary terms are handled on expanding balls (M10.8, M10.9). Pairing (10.15) with χ_R w, χ_R = φ(x/R)^8, leaves flux terms on an annulus where w = v for large R [p. 123]. The transport flux is bounded by powers of ‖φ_R^4 w‖_6, which Sobolev controls by local dissipation (10.17) [pp. 122 to 123]. The pressure flux is nonlocal, so the cutoff is commuted through R_i R_j (10.18): the local part is bounded in L^{3/2}, and the commutator kernel gains min{|x − y|/R, 1}, so its contribution decays like R^{−3/4} (10.19) [pp. 122 to 123]. Every resulting power of the local dissipation is below 2, so Young's inequality absorbs the fluxes, leaving the stretching term ‖∇u‖_∞ E_R plus C_T/R; Gronwall from E_R(0) = 0 gives E_R ≤ C′_T/R, and R → ∞ gives v = u on [0, T] [p. 123]. T < 1 matters: with fixed compact support and unbounded sup norm, the gradient is unbounded (our reading).

Finally (M10.10), the points x_τ = (√(2X_in τ), 0, 0) sit at the similarity point (X_in, 0) and tend to the origin, where c = 1, so (3.6) gives u_θ = τ^{−A}(e_0 + O(τ^{2h})) along them (10.21) [p. 116, p. 124]. A global smooth bounded-energy v equals u on every [0, T], hence on [0, 1), and cannot be bounded near (0, 1) [p. 124].

Every viscosity, and the torus. Rescaling space by √ν gives every viscosity at the same singular time (10.22) [p. 124] (M10.11). For T³, a parabolic dilation with a time shift puts the support inside the open unit cube at the same singular time; integer translates then have disjoint supports with a gap and never interact, and the pressure is periodic, as the erratum to Fefferman's statement requires [p. 125] (M10.12). Periodic uniqueness is simpler: integration over T³ removes the transport and pressure terms [p. 126].

Exactly what is claimed, and in which class

• Constructed at viscosity one: u, p smooth on R³ × [0, 1), divergence-free, supported in K, zero for t ≤ 1 − τ_0 [p. 118]; energy and dissipation at most F(1)² [p. 121]; growth (10.21) [p. 124]. The force is smooth with support in K × [0, 2] [p. 120], is the residual for t < 1, is not claimed divergence-free [p. 118], is nonzero, and depends on ν [p. 124].
• Breakdown class: excluded are smooth solutions on R³ × [0, ∞) with this force, zero datum and uniformly bounded kinetic energy [p. 1, p. 124], with no condition on P or on derivatives of v at infinity [p. 121].
• Lemma 10.5's hypotheses: viscosity one, fixed T < 1, the force of Lemma 10.3, zero initial velocity, v smooth on R³ × [0, T] and in L^∞([0, T]; L²) [p. 121]; general ν follows by inverse rescaling [p. 124]. Nothing is asserted at or after t = 1. The proof uses only an energy bound on each [0, T], less than the stated uniform bound (our reading).
• Lifespan and growth: the maximal classical interval [0, 1) is asserted in the H³ class [p. 124]. Growth is a lower bound along x_τ; no upper bound on ‖u(t)‖_∞ is given (our reading).
• Corollary 10.6: no global smooth periodic solution with the same datum and force; the force is compactly supported in time and the pressure periodic [pp. 125 to 126]. No energy hypothesis is stated; on compact T³, smoothness gives finite energy anyway (our reading).
• Scope: the paper presents these as alternatives (C) and (D) of Fefferman's statement [p. 1, p. 125]. Section 10 assumes Theorem 3.1 and does not address the unforced equations, weak continuations past t = 1, or other data or forces (our reading).

What breaks without each move

• M10.1: curl(cA) may be singular at the axis, and at z = 0, r > 0, ∇c × A would involve A where no bounds exist.
• M10.2: cutting velocities breaks div u = 0; without (10.3) the cutoff could reach q = q∗ or beyond, where nothing is controlled.
• M10.3: cutoff terms away from the origin have no proven limit at t = 1, so f has no smooth extension.
• M10.4: the diverging terms at the singular point have no proven summed limit, and without (10.10) smoothness across t = 1 fails.
• M10.5: f exists only on (0, 1); zero extension jumps and the Taylor series may diverge.
• M10.6: the full field's energy is bounded only heuristically, so Theorem 1.1's energy claim is unproved.
• M10.7: the pressure flux cannot be estimated; a harmonic pressure gradient is not excluded.
• M10.8: the nonlocal flux cannot be absorbed with a remainder decaying in R.
• M10.9: no uniqueness, so the construction says nothing about other solutions.
• M10.10: no proof that the localized field, not only u_loc, diverges, and no contradiction.
• M10.11: the result holds only at viscosity one.
• M10.12: no torus result; overlapping translates would interact.

Three questions the section leaves open

1. Can the force be both divergence-free and compactly supported? Cutting the pressure locally leaves a force with divergence [p. 118]; moving its gradient part into the pressure would in general cost compact support of f and p, since a divergence-free force dictates the Poisson pressure (our reading). The answer would show whether the divergence is an artifact of local cutting or forced by compact support.
2. What do weak solutions with this force do at and after t = 1? Comparison covers only smooth bounded-energy competitors before t = 1 [p. 121], and f past t = 1 is an arbitrary continuation [p. 120]. Leray-type solutions should exist globally (our reading; compare [p. 2]); their singular set, their uniqueness afterward, and the role of the continuation would say what the breakdown means outside the smooth class.
3. How fast does ‖u(t)‖_∞ grow, and where? Only a lower bound at one family of points is proved [p. 124]. Pulse amplitudes scale like q^{−1/2−h/2} [p. 11], below the core scale q^{−A}, so a matching upper bound seems plausible (our reading); it would fix the rate against the parabolic τ^{−1/2}.

Page tags

Pages cited, in order: 1, 2, 6, 7, 11, 15, 16, 17, 33, 49, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 138.

Extraction note: superscripts and subscripts were flattened on pages 116 to 126, and displays on pp. 119, 120 and 123 broke apart; all were recoverable.